A zero-coupon bond with a face value of $1,000 is issued with an initial price of $387.50. The bond matures in 30 years. What is the implicit interest, in dollars, for the first year of the bond's life?

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The implicit interest for the first year can be calculated by finding the interest rate and then applying it to the initial price. The formula for the future value of a present amount at compound interest is $FV = PV(1 + r)^n$, where $FV$ is the future value, $PV$ is the present value, $r$ is the annual interest rate, and $n$ is the number of years. Rearranging the formula to solve for $r$, we get $r = (FV/PV)^{1/n} - 1$. Substituting the given values ($FV = $1,000$, $PV = $387.50$, $n = 30$) into the formula, we can find $r$. However, to find the implicit interest for the first year, we simply need to calculate the interest amount, which is $PV \times r$. Given the initial price of $387.50, and knowing that the bond grows to $1,000 in 30 years, we can calculate the annual growth rate, but the exact rate calculation is not necessary to find the first year's interest if we understand the concept. The interest for the first year is the growth in value from the initial price, which can be approximated without directly calculating the rate by understanding the nature of compound interest and the provided options.The correct approach involves understanding that the bond's value increases exponentially over time, and the interest for the first year is the initial price multiplied by the growth rate. Since we're not directly calculating the growth rate here but rather explaining the correct answer based on the options provided and the initial calculation setup, the key is to recognize that the interest amount is a result of applying the correct growth rate to the initial price.Given the nature of zero-coupon bonds, which pay no interest until maturity and are sold at a deep discount, the interest earned in the first year is implicit in the bond's increasing value. The correct answer, B) $12.44, is derived from understanding that the bond's value in the first year increases by this amount, reflecting the compounded interest that is not paid out but rather added to the bond's value. This is a simplified explanation without the exact calculation of the interest rate, which would require more complex mathematics and is not directly provided in the question. The focus is on understanding the concept of how zero-coupon bonds accrue value over time.

A Zero-Coupon Bond with a Face Value of $1,000 Is